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In this paper, a novel control strategy namely disturbance observer-based control is first applied to stabilization and disturbance rejection for an anti-stable stochastic heat equation with Neumann boundary actuation and unknown boundary external disturbance generated by an exogenous system. A disturbance observer-based boundary control is designed based on the backstepping approach and estimation/cancellation strategy, where the unknown disturbance is estimated in real time by a disturbance observer and rejected in the closed-loop, while the in-domain multiplicative noise whose intensity is within a known finite interval is attenuated. It is shown that the resulting closed-loop system is exponentially stable in the sense of both mean square and almost surely. A numerical example is demonstrated to validate the effectiveness of the proposed control approach.

This paper studies the Mittag-Leffler stabilization for unstable infinite dimensional systems actuated by boundary controllers described by time fractional reaction diffusion equations. Via the Riesz basis method, the stable and the unstable part of the considered system are separated. The Kalman rank criterion, which is a classical linear algebra condition, guarantees the stabilizability of the unstable subsystem. Based on these, a controller governed by a finite dimensional system (so called the finite dimensional controller hereafter) is designed to achieve the Mittag-Leffler stability of the closed loop. From infinite to finite, this methodology is a substantial improvement for the existing control laws.

Patients undergoing percutaneous transforaminal endoscopic discectomy (PTED) often complain of unbearable intraoperative pain. This study is to observe clinical effectiveness and safety of intradiscal local anesthetic injection for intraoperative pain relief.Total 268 patients who underwent PTED were analyzed. Patients were divided into intradiscal saline injection group (group C) and intradiscal local anesthetic injection group (group L). Intradiscal mixture was consisted of saline or local anesthetic + methylene blue, the amount of injected mixture was 3 mL. Demographic data, visual analog scale (VAS) and Quebec Back Pain Disability Scale (QBPDS) scores, mean arterial pressure (MAP) and heart rate (HR), total dosage of fentanyl, satisfaction rate of anesthesia and complications were collected at different timepoints.Compared with group C (3.94 ± 0.57), there was a significant reduction of VAS in group L (2.83 ± 0.28) during fibrous annular operation phase (T2). Group L had a lower total dosage of fentanyl (71 [63, 78] μg) and a higher anesthesia satisfaction rate (95.3%) than group C (82 [70, 132] μg and 73.6%, respectively) (P 0.001). MAP and HR were lower in group L than in group C at T2 (P 0.001). Baseline characteristics and QBPDS scores showed no meaningful intergroup differences. Four cases of complications were reported in this study.Intradiscal local anesthetic injection significantly alleviated intraoperative back pain and increased the satisfaction rate of anesthesia, without severe complications, indicating that this technique is a feasible method for intraoperative back pain relief for patients undergoing PTED.

A linear control system with quadratic cost functional over infinite time horizon is considered without assuming controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function. Classical approaches do not apply for such kind of problems. Existence and non-existence of overtaking optimal controls in various cases are established. Some concrete examples are presented. These results show that the overtaking optimality approach can be used to solve some of the above-mentioned problems and at the same time, the limitation of this approach is also revealed.

The active disturbance rejection control (AD RC), first proposed by Jingqing Han in late 1980s, is a powerful control technology being able to deal with external disturbances and internal uncertainties in large scale for control systems in engineering applications. This survey paper will articulate, from a theoretical perspective, the origin, ideology and progress of ADRC for not only uncertain finite-dimensional systems but also uncertain infinite-dimensional ones. Some recent theoretical developments, general framework and unsolved problems of ADRC for finite-dimensional systems with mismatched disturbances and uncertainties by output feedback, uncertain finite-dimensional stochastic systems, uncertain infinite-dimensional systems described by both the wave equation and the fractional-order partial differential equation are successively addressed, from which we see the challenges and opportunities for this remarkable emerging control technology to various types of control systems.

In this study, we focus on the controllability of fractional-order damped systems in linear and nonlinear cases with multiple time-varying delays in control. For the linear system based on the Mittag-Leffler matrix function, we define a controllability Gramian matrix, which is useful in judging whether the system is controllable or not. Furthermore, in two special cases, we present serval equivalent controllable conditions which are easy to verify. For the nonlinear system, under the controllability of its corresponding linear system, we obtain a sufficient condition on the nonlinear term to ensure that the system is controllable. Finally, two examples are given to illustrate the theory.

We address the output feedback stabilization for a Euler–Bernoulli beam equation with boundary moment control and disturbance. The stabilization of this system has been studied in Guo et al. (J Dyn Control Syst. 2014;20:539–58), where the controller is based on full state feedback. In order to derive the output feedback controller, we design a new disturbance estimator to estimate the total disturbance in the sense that the estimation error signal belongs $L^{2}(0,\infty )$ , and it decays exponentially if the initial state is smooth. Using the estimated total disturbance, we propose a control law to stabilize the system. Using admissibility theory, we show that the closed-loop system is exponentially stable and the signals in the disturbance estimator in the closed-loop are proved to be bounded.

This study aims to address the impulsive controllability of linear fractional impulse controlled systems. The authors establish the Kalman rank criterion to guarantee the impulsive controllability of the considered system. They also present the minimum number of impulsive controls for a realisation of such controllability. An example is finally presented to demonstrate the effectiveness of the authors' results.

This paper investigates performance output tracking for a boundary controlled multidimensional heat equation. It is assumed that the so-called total disturbance (which is composed of internal possibly nonlinear uncertainty and external disturbance) the equation is subject to is on one part of the boundary, and that the control is applied on the rest of the boundary. Using only partial boundary measurement, we first propose an extended state observer to estimate both system state and the total disturbance. This allows us to design a servomechanism and then an output feedback controller. Under the condition that both the reference signal and the disturbance vanish or belong to spaces $H^1(0,\infty ;L^1(\Gamma _1))$ and $L^2(0,\infty ;L^2(\Gamma _0))$ , respectively. We show that the over-all control strategy achieves three objectives on the system performance: first, exponentially output tracking for arbitrary given reference signal; second, uniformly boundedness of all internal signals; and, third, the internal asymptotic stability of the closed-loop system. In addition, the control strategy turns out to be robust to the measurement noise. We provide numerical experiments to illustrate the effectiveness of the proposed control strategy.

This paper presents a parameter-dependent design of feedback compensator with space-varying gains for Mittag-Leffler stabilization of linear time fractional parabolic MIMO partial differential equations subject to space-varying diffusion and reaction coefficients. In the proposed design method, under a boundedness assumption, the reaction coefficient is written in a parametric form. By using the parametric form for the reaction coefficient and multiple non-collocated observation outputs, an observer-based state feedback compensator with space-varying gains is then constructed such that the resulting closed-loop coupled equations are Mittag-Leffler stable. By applying the Lyapunov technique with Caputo fractional derivative and variants of Poincaré–Wirtinger’s inequality, a sufficient condition for the existence of such feedback compensator is presented in terms of standard linear matrix inequalities. Finally, simulation results are presented to support the proposed design method.

In this paper, we investigate the event-triggered boundary feedback control problem for an unstable time fractional wave equation with unknown perturbation at the boundary. To cope with the instability of system when there is no disturbance, the backstepping method is adopted to convert the original unstable system into a stable system. An UDE-based estimator based on low-pass filter is proposed to estimate unknown time-varying input disturbance. With the estimation of disturbance, the event-triggered boundary feedback controller is proposed. It is shown that the event-triggered strategy could asymptotically stabilize system and a positive lower bounded of minimum inter-event time is ensured to exclude the Zeno phenomenon.

This paper considers boundary control of time fractional order diffusion–wave equation. Both boundary stabilization and disturbance rejection are considered. This paper, for the first time, has confirmed, via hybrid symbolic and numerical simulation studies, that the existing two schemes for boundary stabilization and disturbance rejection for (integer order) wave/beam equations are still valid for time fractional order diffusion–wave equations. The problem definition, the hybrid symbolic and numerical simulation techniques, outlines of the methods for boundary stabilization and disturbance rejection are presented together with extensive simulation results. Different dynamic behaviors are revealed for different fractional orders which may stimulate new future research opportunities.

This paper is concerned with the output feedback exponential stabilization for a one-dimensional wave equation with in-domain feedback/recirculation of a boundary velocity with a spatially constant coefficient, which is first studied in [ IEEE Trans. Autom. Control , vol. 62, no. 9, pp. 4760–4767, Sep. 2017]. When there are no boundary internal uncertainty and external disturbance, it is shown that by using one displacement measurement only, the output feedback makes the closed-loop system exponentially stable, which essentially improves the result of [ IEEE Trans. Autom. Control , vol. 62, no. 9, pp. 4760–4767, Sep. 2017]. When there are boundary internal uncertainty and external disturbance, using two displacement measurements only, we present an observer-based output feedback law that contains an infinite-dimensional disturbance estimator used to reject the boundary internal uncertainty and external disturbance. The resulting closed-loop system is shown to be exponentially stable and the state of all subsystem involved are uniformly stable. The Backstepping method for infinite dimensional system and active disturbance rejection control method play important roles in the design.

This paper considers performance output tracking and disturbance rejection for a boundary controlled one-dimensional Euler–Bernoulli beam equation suffered unknown external bounded disturbance. By first proposing a disturbance estimator and then based on this disturbance estimator, we construct a servomechanism to track the reference signal and reject the external disturbance. Four control objectives are achieved: a) the output is exponentially tracking the reference signal; b) the disturbance is rejected; c) all the states of internal-loops are bounded; d) when the disturbance and reference are disconnected, the closed-loop is exponentially stable. Finally, the state of the system is shown to be exponentially tracking the reference state.

In this paper, we pay attention to the boundary feedback stabilization of a one-dimensional flexible string system with the exogenous disturbance that acts on the control end. The active disturbance rejection control (ADRC) and disturbance observer-based control (DOBC) are introduced to reduce the vibration caused by external disturbance. By utilizing Lyapunov method and constructing a suitable Lyapunov functional, we conclude that system is uniformly bounded stable. The numerical simulations are also presented to demonstrate the validity of both methods.

This paper is concerned with the adaptive error feedback regulator problem whereby the 1D Euler--Bernoulli beam equation with general unknown amplitude harmonic disturbance is rejected and general ...

This paper is concerned with Caputo–Hadamard fractional Halanay inequality. A useful fractional derivative inequality regarding on x 2 in the Caputo–Hadamard sense is obtained, which makes Halanay inequality more applicable to choose the Lyapunov function and to study the stability of fractional system.

In this study, the asymptotical stability for several classes of fractional order differential systems with time delay is investigated. The authors first present an integral inequality by which the Halanay inequality is extended to fractional order case. Based on the generalised Halanay inequality, the authors establish several asymptotical stability conditions under which the fractional order systems with time delay are asymptotically stable. It is worth to note that these stability conditions are easy to check without resorting to the solution expression of the systems.

The stability analysis of fractional-order systems with unbounded delay remains an open problem. In this paper, we firstly explore two new integral inequalities. Using these two integral inequalities obtained, the Halanay inequality with unbounded delay is extended to Caputo fractional-order case and Riemann–Liouville fractional-order case. Finally, several examples are presented to illustrate the effectiveness and applicability of the fractional Halanay inequalities in obtaining the asymptotic stability conditions of fractional-order systems with unbounded delay.

This paper concerns a n-dimensional spherically symmetric model for the combustion of a viscous, compressible, radiative-reactive gas with a chemical kinetics equation. Under suitable assum...

In this paper, we are concerned with the output feedback exponential stabilization problem for a system (plant) described by a one-dimensional Euler–Bernoulli beam equation. The measurements are only the displacement and the angular velocity at the right end. An infinite dimensional estimator is designed to estimate the disturbance. With the estimated disturbance, we propose a state observer that is exponentially convergent to the original system, then design two different kinds of stabilizing controllers: one is based on the velocity feedback, the other is based on the angular velocity feedback. In both cases, by adopting the Riesz basis approach, the exponential stability of the closed-loop systems is built with guaranteeing that all internal systems are uniformly bounded. The numerical experiments are carried out to illustrate the theoretical results.

In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in Liang [Nonlinear Dyn. 38 (2004) 339–354] where all results were verified by simulations only.

We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the ...

In this paper, we consider performance output tracking for a boundary controlled one-dimensional wave equation with possibly unknown internal nonlinear uncertainty and external disturbance. We first show that the open-loop system is well-posed and then propose a disturbance estimator. It is shown that the disturbance estimator can estimate successfully the total disturbance that consists of internal uncertainty and external disturbance. An servomechanism based on the estimated total disturbance is then designed. It is shown that the closed-loop system is well-posed. Three control objectives are achieved: (a) the output is tracking the reference signal; (b) all the internal signals are uniformly bounded; (c) the closed-loop system is internally asymptotically stable if both the reference signal and the disturbance vanish or belong to the space H2(0, ∞) and L2(0, ∞), respectively. The unmatched performance output tracking control is first time applied to a system described by the partial differential equation for complete general disturbance rejection and reference tracking purpose. Another key feature of this paper is that we do not use the high-gain to estimate total disturbance for unmatched system. The numerical experiments are carried out to illustrate effectiveness of the proposed control law.

In this paper, we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations. New state feedback controls with actuation on one end are designed by the backstepping method for both Dirichlet and Neumann boundary controls. By the Riesz basis approach and the fractional Lyapunov method, we prove the existence and uniqueness and the Mittag--Leffler stability for the closed-loop systems. For both cases, the observers and the observer-based output feedback are designed to stabilize the systems.

In this paper, we consider boundary output feedback stabilization for a multi-dimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance. A key feature of the proposed observer in this paper is that we do not use the high-gain to estimate the disturbance. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

In this paper, we consider boundary output feedback stabilization for a multidimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance by using the hidden regularity of the wave equation. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

This paper develops systematically the output feedback exponential stabilization for a one-dimensional unstable/anti-stable wave equation where the control boundary suffers from both internal nonlinear uncertainty and external disturbance. Using only two displacement signals, we propose a disturbance estimator that not only can estimate successfully the disturbance in the sense that the error is in L2(0, ∞) but also is free high-gain. With the estimated disturbance, we design a state observer that is exponentially convergent to the state of original system. An observer-based output feedback stabilizing control law is proposed. The disturbance is then canceled in the feedback loop by its approximated value. The closed-loop system is shown to be exponentially stable and it can be guaranteed that all internal signals are uniformly bounded.

We solve the output regulator problem for a system described by a Neumann boundary controlled anti-stable one-dimensional Schrodinger equation by the backstepping approach. The output to be controlled is not required to be measured and its observation operator is assumed to be admissible for a certain Co-semigroup that is related to the original system, while the measurements used for control are located at the boundary First we show that the open-loop system is well-posed. Then we find a state feedback control law to solve the output regulator problem by the backstepping method. We design a finite-dimensional reference observer and an infinite-dimensional disturbance observer. Based on this observer, we propose an output feedback controller to achieve output regulation.

We consider performance output tracking for a boundary controlled multi-dimensional heat equation with possibly unknown internal nonlinear uncertainty and external disturbance on one part and control on the rest of the boundary. The active disturbance rejection control (ADRC) approach is adopted in investigation. With partial boundary measurement only, we propose an extended state observer which is used to estimate both system state and total disturbance, design a servomechanism and an output feedback controller. Three control objectives have been achieved: a) the performance output is exponentially tracking the arbitrary given reference signal; b) all internal signals are uniformly bounded; c) the closed-loop system is internally asymptotically stable whenever both the reference signal and the disturbance vanish or belong to the spaces H1(0, ∞; L1(Γ 1 )) and L2(0, ∞; L2(Γ 0 )), respectively. In addition, when boundary measurement is suffered from the noise, it is shown that the proposed controller is robust to the measurement noise.

In this paper, we propose a new output feedback control law to exponentially stabilize an one-dimensional wave equation with unknown anti-damping coefficient subject to general external disturbance entering the control end. With the proposed control law, although the anti-damping parameter is unknown and the disturbance is presented, it is shown that the resulting closed-loop system still decays exponentially. The numerical experiments are carried out to illustrate effectiveness of the proposed control law.

This work aims to design the disturbance rejection controllers for three classes of fractional heat equations. Based on Filippov’s theory, the existence conclusion for the partial differential inclusion solution (PDIS) is established for fractional heat equations with discontinuous boundary conditions. Boundary control strategies are designed directly without the use of any robust control method to respectively achieve the power-law type stabilization and the asymptotical stabilization for fractional heat equations without and with time delay, respectively. A numerical example is included to illustrate the obtained results.

We investigate the exact controllability of a nonlinear plant described by the equation x˙(t)=Ax(t)+Bu(t)+BNN(x(t),t), where t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group T on a Hilbert space X, B and BN, defined on Hilbert spaces U and UN, respectively, are admissible control operators for T and the function N:X×[0,∞)↦UN is continuous in t and Lipschitz in x, with Lipschitz constant LN independent of t. Thus, B and BN can be unbounded as operators from U and UN to X, in which case the nonlinear term BNN(x(t),t) in the plant is in general not Lipschitz in x. We assume that there exist linear operators F and Fb such that the triples (A,[BBN],F) and (-A,[BBN],Fb) are regular and A + BFΛ and −A + BFb, Λ are generators of operator semigroups Tf and Tb on X such that ∥Ttf∥·∥Ttb∥ decays to zero exponentially. We prove that if LN is sufficiently small, then the nonlinear plant is exactly controllable in some time τ > 0. Our proof is constructive, i.e. given an initial state ...

In this paper, we consider stabilisation for a cascade of ODE and first-order hyperbolic equation with external disturbance flowing to the control end. The active disturbance rejection control (ADR...

The active disturbance rejection control (ADRC) is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics, external disturbance, and unknown part in coefficient of the control. However, all theoretical works up to present are limited to deterministic uncertainty. In this technical note, we generalize the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise. We first design an extended state observer (ESO) that is used to estimate both state, and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance. It is shown that the resulting closed-loop system is practically stable in the mean-square topology. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.

In this paper, we consider output feedback stabilisation for a wave PDE-ODE system with Dirichlet boundary interconnection and external disturbance flowing the control end. We first design a variable structure unknown input type state observer which is shown to be exponentially convergent. Then, we estimate the disturbance in terms of the estimated state, an idea from active disturbance rejection control. These enable us to design an observer-based output feedback stabilising control to this uncertain PDE-ODE system.